Triangular Cross Section Civil Engineering (CE) Notes | EduRev

Advanced Solid Mechanics - Notes, Videos, MCQs & PPTs

Civil Engineering (CE) : Triangular Cross Section Civil Engineering (CE) Notes | EduRev

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Triangular cross section

Finally, we consider the torsion of a cylinder with equilateral triangle cross section, as shown in figure 9.21. The boundary of this section is defined by,

Triangular Cross Section Civil Engineering (CE) Notes | EduRev

where a is a constant and we have simply used the product form of each boundary line equation. Assuming that the Prandtl stress function to be of the form,

Triangular Cross Section Civil Engineering (CE) Notes | EduRev

so that the boundary condition (9.50) is satisfied. It can then be verified that the potential given in equation (9.99) satisfies (9.47) if

Triangular Cross Section Civil Engineering (CE) Notes | EduRev

Substituting equation (9.99) in (9.57) and using (9.100) we obtain the torque to be

Triangular Cross Section Civil Engineering (CE) Notes | EduRev

where the polar moment of inertia for the equilateral triangle section, J = 3 √ 3a4 .

The shear stresses given in equation (9.44) evaluates to

Triangular Cross Section Civil Engineering (CE) Notes | EduRev

 Triangular Cross Section Civil Engineering (CE) Notes | EduRev

on using equations (9.99) and (9.100). The magnitude of the shear stress at any point is given by,

Triangular Cross Section Civil Engineering (CE) Notes | EduRev

Since, for torsion the maximum shear stress occurs at the boundary of the cross section, we investigate the same at the three boundary lines. We begin with the boundary x = a. It is evident from (9.102) that on this boundary σxz = 0. Then, it follows from (9.103) that σyz is maximum at y = 0 and this maximum value is 3aµΩ/2. It can be seen that on the other two boundaries too the maximum shear stress, τmax = 3aµΩ/2. Substituting (9.99) in equations (9.45) and (9.46) and solving the first order differential equations, we obtain the warping displacement as,

Triangular Cross Section Civil Engineering (CE) Notes | EduRev

on using the condition that the origin of the coordinate system does not get displaced; a requirement to prevent the body from displacing as a rigid body.

 

Triangular Cross Section Civil Engineering (CE) Notes | EduRev

Figure 9.22: Example of a multiply connected cross section. Elliptical cross section with two circular holes.

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